# Area versus Perimeter

Discussion in 'General Math' started by anheiserb, Nov 14, 2006.

1. ### anheiserbGuest

Can someone please explain how to answer this question? Thanks....

The perimeter of leaf II is 1.5 times as long as the perimeter of leaf
I. How many times greater is the area of leaf II than the area of
leaf I? Express your answer as a decimal to the nearest hundredth.

Thanks again,
Jeff

anheiserb, Nov 14, 2006

2. ### W H GGuest

The unstated assumption is that the leaves are
geometrically similar. That means that all
lengths are multiplied by the same amount, in
this case given as 1.5. Perimeter is a length so
it is proportional to the length ratio; area depends
on a length squared so it goes up as the length
squared.
Area leaf II / Area leaf I = Perim. of II/ Perim of I

so area ratio = 1.5^2

If you want to do it by baby steps, pretend the leaves
are rectangles and put in some reasonable numbers.

----- W H G

W H G, Nov 14, 2006

3. ### William ElliotGuest

Whew, a clue.
You are making gibberish. Don't you mean
Area leaf II / Area leaf I = (Perim. of II/ Perim of I)^2 ?

William Elliot, Nov 14, 2006
4. ### W H GGuest

8-( right you are.

W H G, Nov 14, 2006
5. ### John BaileyGuest

Not to be a wa but the problem is incompletely specified. The edges
of various leaves have different fractal dimensions.
http://hypertextbook.com/facts/2002/leaves.shtml
"The purpose of this lab is to determine the dimensions of six
different leaves. The dimensions that were obtained ranged from 1.7 to
1.9."
Further, the scale at which the length is measured must be specified.
If a scale of 1mm (for example) is used for both leaves I and II, then
the answer given by other posts (1.5^2) must be increased (a smaller
relative scale is implied in its perimeter measurement--thus a greater
length will result.) I think the increased value would be 2.44 instead
of 2.25 for a maple leaf (fractal dimension 1.9)

I am sorry to confuse the issue. Fractal dimensions are a bit
advanced for the original post's context, but hopefully, it will
stimulate some further exploration of an interesting subject.

http://tinyurl.com/yxxhvn
Fractals in biological Science.
"In fractal geometry, the fractal dimension is a statistical quantity
that gives an indication of how completely a fractal appears to fill
space, as one zooms down to finer and finer scales"

John

John Bailey, Nov 15, 2006
6. ### Roger BagulaGuest

I found this yesterday:
http://mathworld.wolfram.com/IsoperimetricQuotient.html

Roger Bagula, Nov 16, 2006